Science China Information Sciences

, Volume 53, Issue 11, pp 2230–2239 | Cite as

Lattice-valued modal propositional logic and its completeness

Research Papers

Abstract

Based on the concept of the complete lattice satisfying the first and second infinite distributive laws, the present paper introduces the semantics of the lattice-valued modal propositional logic. It is pointed out that this semantics generalizes the semantics of both classical modal propositional logic and [0, 1]-valued modal propositional logic. The definition of the QMR 0-algebra is proposed, and both the Boole-typed latticevalued modal propositional logic system B and the QMR 0-typed lattice-valued modal propositional logic system QML* are constructed by use of Boole-algebras and QMR 0-algebras, respectively. The main results of the paper are the completeness theorems of both the system B and QML*.

Keywords

latticed-valued modal propositional logic modal model QMR0-algebra validity completeness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blackburn P, de Rijke M, Venema Y. Modal Logic. New York: Cambridge University Press, 2001. 1–37MATHGoogle Scholar
  2. 2.
    Cresswell M J, Hughes G E. A New Introduction to Modal Logic. London: Routledge, 1996. 19–39MATHGoogle Scholar
  3. 3.
    Wang G J. Non-classical Mathematical Logic and Approximate Reasoning (in Chinese). 2nd ed. Beijing: Science Press, 2008. 224–251Google Scholar
  4. 4.
    Hájek P. On fuzzy modal logics S5(C). Fuzzy Sets Syst, 2009, doi: 10.1016/j.fss.2009.11.011Google Scholar
  5. 5.
    Mironov A M. Fuzzy modal logics. J Math Sci, 2005, 128: 3641–3483CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ying M S. On standard models of fuzzy modal logics. Fuzzy Sets Syst, 1988, 26: 357–363MATHCrossRefGoogle Scholar
  7. 7.
    Huynh V N, Nakamori Y, Ho T B, et al. A context model for fuzzy concept analysis based upon modal logic. Inf Sci, 2004, 160: 111–129MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Zhang Z Y, Sui Y F, Cao C G, et al. A formal fuzzy reasoning system and reasoning mechanism based on propositional modal logic. Theoret Comput Sci, 2006, 368: 149–160MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Wang G J, Zhou H J. Introduction to Mathematical Logic and Resolution Principle. Beijing: Science Press, Oxford, U.K. Alpha Science International Limited, 2009. 257–323Google Scholar
  10. 10.
    Wang G J. A universal theory of measure and integral on valuation spaces with respect to diverse implication operators. Sci China Ser E, 2000, 43: 586–594MATHMathSciNetGoogle Scholar
  11. 11.
    Wang G J, Zhou H J. Quantitative logic. Inf Sci, 2009, 179: 226–247MATHCrossRefGoogle Scholar
  12. 12.
    Wang G J, Duan Q L. Theory of (n) truth degrees of formulas in modal logic and a consistency theorem. Sci China Ser F-Inf Sci, 2009, 52: 70–83MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chen T Y, Wang D G. The semantics of fuzzy modal propositional logic (in Chinese). J Liaoning Norm Univ (Nat Sci Ed), 2003, 26: 341–343MATHGoogle Scholar
  14. 14.
    Wang D G, Gu Y D, Li H X. Generalized tautology in fuzzy modal propositional logic (in Chinese). Acta Electr Sin, 2003, 35: 261–264Google Scholar
  15. 15.
    Hu M D, Wang G J. Tautologies and quasi-tautologies in fuzzy modal logic (in Chinese). Acta Electr Sin, 2009, 37: 2484–2488Google Scholar
  16. 16.
    Wang G J. The Theory of Topological Molecular Lattices (in Chinese). Xi’an: Shaanxi Normal University Press, 1990. 5–7Google Scholar
  17. 17.
    Davey B A, Priestley H A. Introduction to Lattices and Order. New York: Cambridge University Press, 1990. 114–123MATHGoogle Scholar
  18. 18.
    Erné M. Infinite distributive laws versus local connectedness and compactness properties. Top Appl, 2009, 156: 2054–2069MATHCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Institute of MathematicsShaanxi Normal UniversityXi’anChina

Personalised recommendations